Linear Transformations

大學線性代數再探

大學數學

大 學 線性代數 .

linear operator . 線性代數 , 性 .

linear transformation 性 , 再 . 代數

field 性 over field polynomial ring 代數 (

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Chapter 2

Linear Transformations

學 數學 大 數 性. 數

, 數 數;

group homomorphisms ring homomorphisms. 線性代數 數

vector spaces , linear transformations.

2.1. Definition and Basic Properties

Definition 2.1.1. V,W over F vector spaces. V W 數

T : V → W, v₁, v₂∈ V r∈ F T (rv₁+ v₂) = rT (v₁) + T (v₂), T linear transformation ( linear mapping) from V to W .

T is F-linear. L (V,W) V W linear

transformations .

Question 2.1. T is F-linear v, v^′∈ V r∈ F T (v + v^′) = T (v) + T (v^′) T (rv) = rT (v) ?

linear transformation 性 , linear transforma-

tion vector spaces, O_V V .

Proposition 2.1.2. T : V → W linear transformation, (1) T (O_V) = O_W

(2) v∈ V T (−v) = −T(v).

Proof.

(1) T (OV) = T (OV+ OV) = T (OV) + T (OV), T (OV) = OW.

(2) v∈ V, O_W = T (v + (−v)) = T(v) + T(−v) T (−v) = −T(v).

 23

linear transformations linear transformation. T, T^′

V W linear transformations, V W 數 T + T^′: V → W

v∈ V, (T + T^′)(v) = T (v) + T^′(v). r∈ F , V W

數 rT : V → W v∈ V, (rT)(v) = rT(v). 數 linear

transformation.

Proposition 2.1.3. T, T^′ V W F-linear transformations r∈ F , T + T^′ rT V W F-linear transformations.

Proof. v₁, v₂∈ V s∈ F, (T + T^′)(sv₁+ v₂) = T (sv₁1 + v₂) + T^′(sv₁+ v₂) T, T^′ F-linear, T (sv1+ v₂) + T^′(sv₁+ v₂) = sT (v₁) + T (v₂) + sT^′(v₁) + T^′(v₂) = s(T (v1) + T^′(v1)) + (T (v2) + T^′(v2)). (T + T^′)(sv1+ v2) = s(T + T^′)(v1) + (T + T^′)(v2).

(rT )(sv₁+ v₂) = rT (sv₁+ v₂) = rsT (v₁) + rT (v₂) = s(rT (v₁)) + rT (v₂) = s(rT )(v₁) +

(rT )(v2). 

Question 2.2. V W linear transformations L (V,W),

Proposition 2.1.3 L (V,W) vector space over F?

數 數 , 數.

Proposition linear transformations linear transformation.

Proposition 2.1.4. T₁: V → W, T2 : W → U F-linear, T₂◦ T1: V → U F-linear.

Proof. v, v^′ ∈ V r∈ F, T2◦ T1(rv + v^′) = T2(T1(rv + v^′)). T1 F- linear, T₁(rv + v^′) = rT₁(v) + T₁(v^′)再 T₂ F-linear T₂◦T1(rv + v^′) = T + 2(rT₁(v) + T1(v^′)) = rT₂(T₁(v)) + T₂(T₁(v^′)) = rT₂◦ T1(v) + T₂◦ T1(v^′).  Question 2.3. T, T^′ V W F-linear transformations, T^′′ W U F-linear transformation. T^′′◦ (T + T^′) = T^′′◦ T + T^′′◦ T^′? r∈ F r(T^′′◦ T) = (rT^′′)◦ T = T^′′◦ (rT)?

F-spaces V,W , V W linear transformation.

Theorem V W linear transformations .

Theorem 2.1.5. {v1, . . . , v_n} V basis, w₁, . . . , w_n∈ W, F-linear transformation T : V → W T (vi) = wi, ∀i ∈ {1,...,n}.

Proof. 性: T ∈ L (V,W) T (v_i) = w_i. T : V →

W , v = c₁v₁+···+cnv_n∈ V, T(v) = c1w₁+···+cnw_n. {v1, . . . , v_n} V basis, T V W well-defined function. T T (vi) = wi,

T F-linear. v =∑ⁿi=1c_iv_i, v^′=∑ⁿi=1d_iv_i∈ V r∈ F, T (rv + v^′) =

2.2. Image and Kernel 25

T (∑ⁿi=1(rc_i+ d_i)v_i) =∑ⁿi=1(rc_i+ d_i)w_i; rT (v) + T (v^′) = rT (∑ⁿi=1c_iv_i) + T (∑ⁿi=1d_iv_i) = r∑ⁿi=1ciw_i+∑ⁿi=1diw_i, vector space 性 , T (rv + v^′) = rT (v) + T (v^′).

性: T^′: V → W F-linear T^′(v_i) = w_i,∀i ∈ {1,...,n}, T = T^′. T (v) = T^′(v),∀v ∈ V. v =∑ⁿi=1civ_i∈ V, T T (v) =

∑ⁿi=1c_iw_i, T^′ F-linear T^′(v) ∑ⁿi=1c_iT (v_i) =∑ⁿi=1c_iw_i, T = T^′.  Theorem 2.1.5 , linear transformation T : V → W,

basis S u∈ S, T(u) , v∈ V, T (v) !

T_θ :R² → R² R² (x, y) (0, 0) θ

. T_θ((x, y)) ? (1, 0) = (cos 0, sin 0) (0, 0) θ

(cosθ,sinθ), T_θ((1, 0)) = (cosθ,sinθ). (0, 1) = (cos(π/2),sin(π/2)), T_θ((0, 1)) = (cos((π/2)+θ),sin((π/2)+θ)) = (−sinθ,cosθ).

T_θ((x, y)) = T_θ(x(1, 0) + y(0, 1)) = xT_θ((1, 0)) + yT_θ((0, 1)) = x(cosθ,sinθ) + y(−sinθ,cosθ) = (x cosθ − ysinθ,xsinθ + ycosθ). , linear transformation ,

數 , 數 linear transformation .

, T_θ linear transformation ( ),

T_θ((x, y)) = (x cosθ − ysinθ,xsinθ + ycosθ).

Question 2.4. T :R² → R² linear transformation, T ((1, 2)) = (2, 1), T ((2, 4)) = (4, 2) T ((x, y)) = (y, x)? T^′:R²→ R² linear transfor- mation, T^′((1, 2)) = (2, 1), T^′((2, 1)) = (1, 2) T^′((x, y)) = (y, x)?

2.2. Image and Kernel

Linear transformation vector spaces , “ ”

subspaces. , 數 f : S1→ S2. S^′₁⊆ S1,

f (S^′₁) ={ f (s) | s ∈ S^′1}.

f (S^′₁) S2 subset, the image of S^′₁ under f ; S₂^′ ⊆ S2, f⁻¹(S₂^′) ={s ∈ S1| f (s) ∈ S^′₂}.

f⁻¹(S^′₂) S₁ subset, the preimage of S^′₂ under f .

Question 2.5. Image preimage inclusion-preserving? 數 f : S₁→ S₂, S^′′₁⊆ S^′₁⊆ S1 f (S^′′₁)⊆ f (S^′₁)? S^′′₂⊆ S^′₂⊆ S2, f⁻¹(S^′′₂)⊆ f⁻¹(S^′₂)?

Question 2.6. f : S₁→ S2 數, S^′₁, S^′′₁⊆ S1 S^′₂, S^′′₂⊆ S2.

?

(1) f (S^′₁∩ S^′′₁) = f (S^′₁)∩ f (S^′′₁).

(2) f (S^′₁∪ S^′′₁) = f (S^′₁)∪ f (S^′′₁).

(3) f⁻¹(S^′₂∩ S^′′₂) = f⁻¹(S^′₂)∩ f⁻¹(S^′′₂).

(4) f⁻¹(S^′₂∪ S^′′₂) = f⁻¹(S^′₂)∪ f⁻¹(S^′′₂).

linear transformation subspace 性 . Lemma 2.2.1. T : V → W linear transformation.

(1) V^′ V subspace, T (V^′) W subspace.

(2) W^′ W subspace, T⁻¹(W^′) V subspace.

Proof. T (V^′)⊆ W T⁻¹(W^′)⊆ V, Proposition 1.2.1 . (1) O_V ∈ V^′ ( V^′ subspace), Proposition 2.1.2 (1) O_W = T (OV)∈

T (V^′). 再 w₁, w₂ ∈ T(V^′) r, s∈ F , v₁, v₂∈ V^′ w₁= T (v1), w2= T (v2). v = rv₁+ sv2∈ V^′, rw1+ sw2= T (v)∈ T(V^′),

T (V^′) W subspace.

(2) O_W ∈ W^′ T (OV) = OW ∈ W^′ O_V ∈ T⁻¹(W^′). 再 v₁, v2 ∈ T⁻¹(W^′) r, s∈ F, T (v₁)∈ W^′ T (v₂)∈ W^′ . W^′ W subspace

T (rv1+ sv2) = rT (v1) + sT (v2)∈ W^′, rv1+ sv2∈ T⁻¹(W^′), T⁻¹(W^′) V subspace.



, V^′= V W^′={OW} , .

Definition 2.2.2. T : V → W linear transformation.

(1) T (V ) the image (or range) of T , Im(T ) .

(2) T⁻¹({OW}) the kernel (or null-space) of T , Ker(T ) . Lemma 2.2.1 Im(T ) W subspace, Ker(T ) V subspace.

Question 2.7. image preimage inclusion-preserving Im(T ) = T (V ) subspaces image 大 subspace, Ker(T ) = T⁻¹({OW}) subspaces

preimage . 探 T ({OV}) T⁻¹(W ) ?

數, 數 (onto) (one-to-one).

Im(T ) Ker(T ) T linear transformation , Im(T )

Ker(T ) T , .

Proposition 2.2.3. T : V → W linear transformation.

(1) T Im(T ) = W .

(2) T Ker(T ) ={OV}.

Proof. Im(T )⊆ W {OV} ⊆ Ker(T), (1) W⊆ Im(T)

, (2) Ker(T )⊆ {OV} .

2.2. Image and Kernel 27

(1) T onto, w∈ W, v∈ V T (v) = w, w∈ T(V) = Im(T).

W ⊆ Im(T), W = Im(T ). , W ⊆ Im(T) w∈ W

Im(T ) , v∈ V w = T (v), T onto.

(2) T one-to-one, V T (v) = O_W v O_V ( T (O_V) =

O_W). v∈ Ker(T), T (v) = OW, v = O_V. Ker(T ) ={OV}.

, Ker(T ) ={OV}. v₁, v₂∈ V T (v₁) = T (v₂), T linear T (v1−v2) = O_W, v₁−v2∈ Ker(T) = {OV}. v₁= v₂, T one-to-one.



Proposition 2.2.3 , T linear T onto

Im(T ) = W ; T one-to-one Ker(T ) ={OV} T

linear . 數 f , f linear f⁻¹({0}) = {0}

f one-to-one.

image kernel , . Lemma

image spanning set kernel linear independency .

Lemma 2.2.4. T : V → W linear transformation, S, S^′ V subsets.

(1) S V spanning set, T (S) Im(T ) spanning set.

(2) S^′ linearly independent Span(S^′)∩ Ker(T) = {OV}, T (S^′) linearly independent.

Proof.

(1) w∈ Im(T) v∈ V w = T (v), V = Span(S)

c1, . . . , c_n∈ F v₁, . . . , v_n∈ S v = c₁v₁+···+cnv_n. T (v1), . . . , T (v_n)∈ T (S), T linear w = T (c₁v₁+··· + cnv_n) = c₁T (v₁) +··· + cnT (v_n)∈ Span(T (S)). Im(T )⊆ Span(T(S)). , w∈ Span(T(S)),

c1, . . . , cn∈ F w₁, . . . , wn∈ T(S) w =∑ⁿi=1ciw_i. w_i∈ T(S)

v_i ∈ S w_i= T (v_i), w =∑ⁿi=1c_iT (v_i) =∑ⁿi=1T (c_iv_i)∈ T(Span(S)) = T (V ) = Im(T ), Span(T (S))⊆ Im(T).

(2) T (S^′) . v, v^′∈ S^′ v, v^′

T (v) = T (v^′), T (v− v^′) = O_W, v− v^′∈ Ker(T). v− v^′∈ Span(S^′), v− v^′∈ Span(S^′)∩ Ker(T) = {OV} v = v^′ . T (S^′) .

T (S^′) linearly dependent, v₁, . . . , v_n∈ S^′ c₁, ..., c_n∈ F 0 c₁T (v₁) +··· + cnT (v_n) = O_W. T linear T (c₁v₁+··· + cnv_n) = O_W, c₁v₁+··· + cnv_n∈ Ker(T). c₁v₁+··· + cnv_n∈ Span(S^′), Span(S^′)∩ Ker(T) = {OV}, c1v₁+··· + cnv_n= OV. S^′ linearly independent , T (S^′) linearly independent.



Lemma 2.2.4 (1) T : V → W linear transformation, V finite dimensional F-space, Im(T ) finite dimensional F-space.

Question 2.8. T : V→ W linear transformation. V finite dimensional F-space, dim(Im(T ))≤ dim(V)? dim(W ) > dim(V ) T

?

V basis Im(T ) basis.

Theorem 2.2.5. T : V → W linear transformation. S₀ Ker(T ) basis S0∪ S V basis, T (S) Im(T ) basis.

Proof. T (S) Im(T ) spanning set: S0∪ S V spanning set, Lemma 2.2.4 (1) T (S₀∪S) Im(T ) spanning set. S₀∈ Ker(T) T (S₀) ={OW}.

T (S0∪ S) = T(S0)∪ T(S) = {OW} ∪ T(S), {OW} ⊆ Span(T(S) ( Corollary 1.3.4) Span(T (S)) = Span({OW} ∪ T(S)) = Span(T(S0∪ S)) = Im(T).

T (S) linearly independent: S0∪ S linearly independent S0 linearly independent, Corollary 1.4.4 S linearly independent Span(S)∩Ker(T) = Span(S)∩

Span(S0) ={OV}. Lemma 2.2.4 (2) T (S) linearly independent.  , V finite dimensional vector space, V , Ker(T ) Im(T )

dimension .

Corollary 2.2.6 (Dimension Theorem). V finite dimensional F-space T : V → W linear transformation,

dim(V ) = dim(Ker(T )) + dim(Im(T )).

Proof. Theorem 1.5.8 Ker(T ) basis S₀ ={v1, . . . , v_m}, 再 Theorem 1.5.9 S ={vm+1, . . . , vn} S0∪ S = {v1, . . . , vm, vm+1, . . . , vn} V

basis. Theorem 2.2.5 {T(vm+1), . . . , T (v_n)} Im(T ) basis, n− m = dim(Im(T)), dim(V ) = n = m + (n− m) = dim(Ker(T)) + dim(Im(T)). 

V finite dimensional vector space T : V → W linear transformation, dim(Im(T )) the rank of T , dim(Ker(T )) the nullity of T . Dimension Theorem Rank Theorem: rank of T + nullity of T = dim(V ).

Question 2.9. T : V→ W linear transformation. V finite dimensional

F-space, dim(W ) < dim(V ) T ?

2.3. Isomorphism 29

2.3. Isomorphism

Linear transformation vector spaces . vector spaces linear transformation, vector spaces

, isomorphic. 探 isomorphism 性

.

Definition 2.3.1. T : V→ W linear transformation. T one-to-one and

onto, T isomorphism. V W isomorphic V ≃ W .

Proposition 2.2.3 T : V → W isomorphism Im(T ) = W and

Ker(T ) ={OV}. T isomorphism , S V basis, T one-to-

one T (S) ( v, v^′ ∈ S v̸= v^′, T (v)̸= T(v^′)), 再 Lemma

2.2.4 T (S) W basis. , T (S) T (S) W

basis, Span(T (S)) = W , Im(T ) = W . , v∈ Ker(T), S V basis, v₁, . . . , v_n∈ S , c₁, . . . , c_n∈ F v = c₁v₁+··· + cnv_n. O_W = T (c1v₁+···+cnv_n) = c₁T (v1) +···+cn(T v_n). T (S) basis T (v1), . . . , T (v_n)∈ T(S) c1, . . . , cn 0, v = O_V, Ker(T ) ={OV}. , . Proposition 2.3.2. T : V→ W linear transformation S V basis.

T isomorphism T (S) T (S) W basis.

Question 2.10. Proposition 2.3.2 T (S) ?

V finite dimensional vector space , .

Corollary 2.3.3. V,W F-spaces V finite dimensional F-space. V ≃ W dim(V ) = dim(W ).

Proof. V≃ W isomorphism T : V → W, Proposition 2.3.2 W finite dimensional F-space dim(W ) = dim(V ). , W finite dimensional F-space dim(W ) = dim(V ) = n, V basis{v1, . . . , vn} W basis{w1, . . . , wn},

Theorem 2.1.5 linear transformation T : V → W T (v_i) = w_i, ∀i ∈ {1,...,n}. {T(v1) . . . , T (v_n)} = {w1, . . . , w_n} W basis, Proposition 2.3.2

T isomorphism, V ≃ W. 

Corollary 2.3.3 finite dimensional vector spaces isomorphic

, dimension . isomorphism

, finite dimension dimension .

, infinite dimensional vector space,

linear transformation 性 isomorphism vector spaces

, finite dimensional , dimension

( 大 dimension 性).

數 f , 數 f^◦−1 , f linear transformation 數 f^◦−1 linear transformation?

. , preimage , f^◦−1 f

數.

Proposition 2.3.4. T : V → W isomorphism, T inverse ( 數) T^◦−1: W→ V isomorphism.

Proof. T one-to-one and onto, inverse T^◦−1 one-to-one and onto.

T^◦−1 W V linear transformation . w, w^′∈ W, T isomorphism v, v^′ ∈ V T (v) = w T (v^′) = w^′. inverse T^◦−1(w) = v T^◦−1(w^′) = v^′. r∈ F, T^◦−1 linear transformation,

T^◦−1(rw + w^′) = rT^◦−1(w) + T^◦−1(w^′) = rv + v^′. T linear, T (rv + v^′) = rT (v) + T (v^′) = rw + w^′. 再 inverse T^◦−1(rw + w^′) = rv + v^′, T^◦−1: W→ V

linear transformation, isomorphism. 

Question 2.11. Vector spaces isomorphic equivalent relation?

Question 2.12. V finite dimensional vector space, Theorem 2.1.5 Proposition 2.3.4 ?

大 學 代數 Isomorphism Theorems.

linear transformation linear transformation . T : V → W linear transformation U Ker(T ) subspace, T : V /U→ Im(T) T (v) = T (v),∀v ∈V/U. T well-defined function.

, v₁= v2 in V /U, T (v1) = T (v2). v₁= v2, v₁− v2∈ U ⊆ Ker(T), T (v₁− v2) = O_W, T (v₁) = T (v₁) = T (v₂) = T (v₂). , v₁, v₂∈ V/U r∈ F,

T (rv₁+ v₂) = T (rv₁+ v₂) = T (rv₁+ v₂) = rT (v₁) + T (v₂) = rT (v₁) + T (v₂), T linear transformation.

Theorem 2.3.5. T : V→ W linear transformation U Ker(T ) sub- space, 數 T : V /U→ Im(T) T (v) = T (v),∀v ∈ V/U linear transformation

Ker(T ) = Ker(T )/U Im(T ) = Im(T ).

Proof. 前 , T linear transformation. v∈ Ker(T), O_W = T (v) = T (v), v∈ Ker(T), v∈ Ker(T)/U. , v∈ Ker(T)/U, v∈ Ker(T), T (v) = T (v) = OW, v∈ Ker(T). Ker(T ) = Ker(T )/U . ,

w∈ Im(T) v∈ V/U w = T (v) = T (v) w∈ Im(T).

Im(T ) = Im(T ). 

U = Ker(T ), .

2.3. Isomorphism 31

Corollary 2.3.6 (The First Isomorphism Theorem). T : V→ W linear trans- formation T : V /Ker(T )→ Im(T) T (v) = T (v), T isomorphism,

V /Ker(T )≃ Im(T).

Proof. T : V /Ker(T )→ Im(T) linear transformation Ker(T ) = Ker(T )/Ker(T ) = {OV} Im(T ) = Im(T ), T isomorphism V /Ker(T )≃ Im(T).  Question 2.13. V finite dimensional vector space, quotient space dimension 性 Dimension Theorem V /Ker(T )≃ Im(T) ?

Theorem 2.3.6 The First Isomorphism Theorem, Isomorphism Theorems.

Corollary 2.3.7 (The Second Isomorphism Theorem). V vector space U,W V subspaces.

(U +W )/U ≃ W/U ∩W.

Proof. U U + W subspace, (U + W )/U vector space.

T : W→ (U +W)/U T (w) = w,∀w ∈ W. T linear transformation.

Im(T ) = (U + W )/U Ker(T ) = U∩W, Corollary 2.3.6 (U +W )/U≃ W/U ∩W.

Im(T )⊆ (U +W)/U, Im(T )⊇ (U +W)/U. v∈ (U +W)/U,

u∈ U,w ∈ W v = u + w. T (u) = u, u = v.

v− (u + w) ∈ W ( v = u + w) w∈ W, v− u = v − (u + w) + w ∈ W, v = u = T (u)∈ Im(T). Im(T ) = (U +W )/U .

u∈ Ker(T), T U , u∈ U. (U +W )/U

O, O V , O = T (u) = u. u = u−O ∈ W, u∈ U ∩W,

Ker(T )⊆ U ∩W. , u∈ U ∩W u∈ W, u = O. T (u) = u = O, u∈ Ker(T),

Ker(T ) = U∩W. 

, linear transformation the First

Isomorphism Theorem, isomorphic 性 .

Corollary 2.3.8 (The Third Isomorphism Theorem). V vector space U,W V subspaces U⊆ W.

(V /U )/(W /U )≃ V/W.

Proof. , U⊆ W ⊆ V vector spaces, V /U V /W vector

spaces. T : V → V/W, T (v) = v, ∀v ∈ V. Ker(T ) = W

Im(T ) = V /W . U ⊆ Ker(T) = W, Theorem 2.3.5, T : V /U → V/W

linear transformation, Im(T ) = Im(T ) = V /W Ker(T ) = Ker(T )/U = W /U .

Corollary 2.3.6 (V /U )/Ker(T )≃ Im(T), (V /U )/(W /U )≃ V/W.  Question 2.14. V finite dimensional vector space, dimension the Second and Third Isomorphism Theorems ?

Question 2.15. V finite dimensional vector space U,W V subspaces U⊆ W ⊆ V. dim(V /U ) = dim(V )− dim(U),dim(V/W) = dim(V) − dim(W) dim(W /U ) = dim(W )− dim(U). dim(V /U )− dim(V/W) = dim(W/U),

(V /U )/(V /W )≃ W/U?

2.4. The Matrix Connection

vector space V , basis , V basis

, V . , linear

transformation, vector space basis, .

linear transformation matrix . 大 前 over R over C

matrix. matrix 性 , over field ,

大 性 , 再 .

V finite dimensional vector space V basis{v1, . . . , vn},

v_i , β = (v1, . . . , v_n) basis, V

ordered basis. , basis , ordered basis,

, , ordered basis.

β = (v1, . . . , v_n),β^′= (v^′₁, . . . , v^′_n) V ordered bases, β = β^′ v_i= v^′_i,

∀i = 1,...,n.

dim(V ) = n, V ordered basisβ , V Fⁿ

linear transformationτβ : V→ Fⁿ, v∈ V, β v v = c₁v₁+···+cnv_n,

τβ(v) =



 c1

... cn



.

Fⁿ column vector, (c₁, . . . , c_n)^t (

row vector (c1, . . . , cn) ). Fⁿ column vector 性 ,

. β ordered basis, τβ well-defined,

isomorphism. 言 , V , τ_β Fⁿ

column vector. Fⁿ column vector, τ_β^◦−1 V

. ordered basis , ordered basis

V Fⁿ column vector vector space .

, V Fⁿ τβ(v) (

2.4. The Matrix Connection 33

), Fⁿ column vector (c₁, . . . , c_n)^t V τ_β^◦−1((c₁, . . . , c_n)^t) , c1v₁+··· + cnv_n.

Example 2.4.1. P2(R) = {ax²+ bx + c| a,b,c ∈ R} R-space,

ordered basis β = (x², x + 1,−1). ax²+ bx + c = a(x²) + b(x + 1) + (b− c)(−1), τ_β(ax²+ bx + c) = (a, b, b− c)^t. τ_β(x²+ x + 1) = (1, 1, 0)^t,

τ_β^◦−1((1, 1, 0)^t) = 1(x²) + 1(x + 1) + 0(−1) = x²+ x + 1.

P₁(R) = {ax + b | a,b ∈ R} R-space, ordered basis β^′= (x− 1,x + 1). ax + b = r(x− 1) + s(x + 1), r = (a− b)/2,s = (a + b)/2, τ_β^′(ax + b) = ((a− b)/2,(a + b)/2)^t.

Question 2.16. Example 2.4.1 β,β^′ β = (−1,x + 1,x²),β^′= (x + 1, x− 1), τ_β(ax²+ bx + c),τ_β^′(ax + b) ?

linear transformation T : V → W, V,W ordered basis β = (v1, . . . , vn),β^′= (w1, . . . , wm). β,β^′, T over F m×n (m row,

n column) . column : i column

τ_β^′(T (vi)). T (vi) β^′ ordered basis T (vi) = c1w₁+···+cmw_m,

i-th column



 c₁

... c_m



. T β,β^′ , _β′[T ]_β

,

β^′[T ]_β=

(τβ^′(T (v₁)), . . . ,τβ^′(T (v_n)) )

,

τ_β^′(T (v_i))∈ F^m,∀i = 1,...,n m×1 column vector, _β′[T ]_β m× n over F matrix.

Example 2.4.2. Example 2.4.1, P₂(R) ordered basisβ = (x², x + 1,−1), P1(R) ordered basisβ^′= (x− 1,x + 1). T : P2(R) → P1(R)

T (ax²+ bx + c) = 2ax + b,

T linear transformation. dim(P₁(R)) = 2, dim(P2(R)) = 3

β^′[T ]_β 2× 3 matrix. T (x²) = 2x, T (x + 1) = 1, T (−1) = 0, Example 2.4.1

τβ^′(T (x²)) = ( 1

1 )

,τβ^′(T (x + 1)) = ( ₋₁

21 2

)

,τβ^′(T (−1)) = ( 0

0 )

,

β^′[T ]_β =

( 1 ⁻¹₂ 0 1 ¹₂ 0

) .

Question 2.17. Example 2.4.2 β,β^′ β = (−1,x + 1,x²),β^′= (x + 1, x− 1),

β^′[T ]_β ?

β^′[T ]_β ? the representative matrix of T with respect to

β,β^′. _β′[T ]_β 代 T linear transformation. ,

m× n over F matrix A, Fⁿ F^m 數 m_A: Fⁿ→ F^m,

Fⁿ column vector x A m× n matrix, A· x F^m

column vector, mA(x) = A· x. 性 A· (rx + x^′) = rA· x + A · x^′,

m_A: Fⁿ→ F^m linear transformation. _β′[T ]_β, Fⁿ

F^m linear transformation m_{β′[T ]β} : Fⁿ→ F^m, T : V → W , :

V T _-

W

v ^- T (v)

τβ(v)

? - _β′[T ]_β·τβ(v) 6

Fⁿ τβ

?

mβ^′[T ]_β

- F^m τ_β^◦−1′

6

v∈ V τβ Fⁿ column vector τβ(v), τβ(v)

m× n matrix_β′[T ]_β, _β′[T ]_β·τ_β(v) F^m column vector.

τ_β^′: W → F^m 數 τ_β^◦−1′ : F^m→ W _β^′[T ]_β·τ_β(v) W τ_β^◦−1′ (_β′[T ]_β·τβ(v)).

T (v). , T τ_β^◦−1′ ◦ m_β′[T ]_β◦τ_β 數,

, T (v) , .

Example 2.4.3. matmul Examples 2.4.1 2.4.2,

T (ax²+ bx + c) = 2ax + b. P₂(R) ax²+ bx + c R³ τ_β(ax²+ bx + c) = (a, b, b− c)^t, 再 _β′[T ]_β

( 1 ⁻¹₂ 0 1 ¹₂ 0

)

·



 a

b b− c



 =(

a−¹₂b a +¹₂b

) .

R² P1(R) (a− (b/2))(x − 1) + (a + (b/2))(x + 1) = 2ax + b T (ax²+ bx + c) .

T =τ_β^◦−1′ ◦ m_β′[T ]_β◦τβ 前, .

m× n matrix A, Fⁿ column vector x, A· x F^m column vector.

, A₁, . . . , A_n A 1 n column ( A n column

2.4. The Matrix Connection 35

column F^m column vector), x = (x₁, . . . , x_n)^t, A· x = x1A₁+··· + xnA_n.

T τ_β^◦−1′ ◦ m_β′[T ]_β◦τβ V→ W linear transformation, Theorem 2.1.5,

, β basis v_i, 代 .

τβ(v_i) = (0, . . . , 1, . . . , 0)^t, (0, . . . , 1, . . . , 0) i 1

0. 前 _β′[T ]_β·τ_β(vi) =_β′[T ]_β· (0,...,1,...,0)^t _β^′[T ]_β i column, 前 column τβ^′(T (v_i)),

τ_β^◦−1′ ◦ m_β′[T ]_β◦τ_β(vi) =τ_β^◦−1′ (_β′[T ]_β·τ_β(vi)) =τ_β^◦−1′ (τ_β^′(T (vi))) = T (vi),∀i = 1,...,n

T =τ_β^◦−1′ ◦ m_β′[T ]_β◦τ_β. (2.1)

V,W vector spaces, V W linear transformation

vector space, L (V,W) . Mm×n(F) over F m×n matrices.

性 , M_m×n(F) vector space. V ordered basis

β = (v1, . . . , vn) W ordered basis β^′= (w1, . . . , wm), 前 linear transformation

matrix , L (V,W) M_m×n(F) 數 Φ,

linear transformation T : V→ W _β^′[T ]_β m×n matrix, Φ(T) = _β^′[T ]_β,∀T ∈

L (V,W). Φ linear transformation, T1, T2∈ L (V,W)

r∈ F, Φ(rT1+ T₂) =_β′[rT₁+ T₂]_β rΦ(T1) +Φ(T2) = r(_β′[T₁]_β) +_β′[T₂]_β .

, _β′[rT1+ T2]_β i-th column τ_β^′((rT1+ T2)(vi)) =τ_β^′(rT1(vi) + T2(vi)), τ_β^′ linear, rτβ^′(T₁(v_i)) +τβ^′(T₂(v_i)). _β′[T₁]_β,_β′[T₂]_β i-th column

τ_β^′(T₁(v_i)),τ_β^′(T₂(v_i)), 數 r(_β′[T₁]_β) +_β′[T₂]_β i-th column rτ_β^′(T1(vi)) +τ_β^′(T2(vi)). _β′[rT1+ T2]_β r(_β′[T1]_β) +_β′[T2]_β , Φ linear transformation.

Φ : L (V,W) → Mm×n(F) isomorphism ( one-to-one and onto).

A∈ Mm×n(F), A i-th column A_i, τ_β^◦−1′ (A_i)∈ W. Theorem 2.1.5 linear transformation T : V → W T (v_i) =τ_β^◦−1′ (A_i),∀i = 1,...,n. ,

β^′[T ]_β i-th row

τ_β^′(T (v_i)) =τ_β^′(τ_β^◦−1′ (A_i)) = A_i,

β^′[T ]_β= A. T L (V,W) Φ(T) =_β^′[T ]_β= A linear transformation,

Φ isomorphism. .

Theorem 2.4.4. V,W vector spaces dim(V ) = n, dim(W ) = m. V,W ordered basis β,β^′. Φ : L (V,W) → Mm×n(F) Φ(T) =_β^′[T ]_β,∀T ∈ L (V,W), Φ

isomorphism,

L (V,W) ≃ Mm×n(F).

Question 2.18. dim(V ) = n, dim(W ) = m, Theorem 2.4.4 dim(L (V,W)) = dim(Mm×n(F)) = mn, Mm×n(F) , L(V,W ) basis?

Question 2.19. A∈ Mm×n(F). m_A : Fⁿ→ F^m m_A(x) = A· x,∀x ∈ Fⁿ. Im(mA) ={A · x ∈ F^m| x ∈ Fⁿ} A column space, C(A) , dim(C(A)) the rank of A. Ker(m_A) ={x ∈ Fⁿ| A · x = O} A null space, N(A) , dim(N(A)) the nullity of A. T : V→ W linear transformation Φ(T) = A, τ_β,τ_β^′ isomorphism, Ker(T )≃ N(A) Im(T )≃ C(A).

rank of A + nullity of A = n?

前 linear transformation T matrix _β′[T ]_β .

V T _-

W

Fⁿ τβ

? τ_β^◦−1 6

mβ^′[T ]_β

- F^m τβ^′

? τ_β^◦−1′

6

T =τ_β^◦−1′ ◦ m_β′[T ]_β◦τ_β, commutative diagram.

Commutative diagram , , 數

. V W : T ; V τ_β Fⁿ,

再 m_{β′[T ]β} F^m, τ_β^◦−1′ W . commutative diagram

T =τ_β^◦−1′ ◦ m_β′[T ]_β◦τ_β. 性, F^m W τ_β^◦−1′

isomorphism W F^m , 數 τ_β^′. V F^m

: T V W , 再 τβ^′ F^m; τβ V Fⁿ,

再 m_{β′[T ]β} F^m. τ_β^′◦ T = m_β′[T ]_β◦τ_β.

τβ^′◦ T =τβ^′◦ (τ_β^◦−1′ ◦ m_β′[T ]_β◦τβ) = (τβ^′◦τ_β^◦−1′ )◦ m_β′[T ]_β◦τβ= m

β^′[T ]_β◦τβ,

commutative diagram 數 .

V Fⁿ , T V W τ_β^′

F^m F^m Fⁿ ( _β′[T ]_β invertible).

Question 2.20. Fⁿ F^m ? 代 數 ?

commutative diagram 數 . V,W,U

vector spaces, dim(V ) = n, dim(W ) = m, dim(U ) = q β,β^′β^′′ ordered bases. T₁: V → W, T2: V → W linear transformations commutative